Finite normal implies potentially characteristic: Difference between revisions

Latest revision as of 06:59, 22 February 2013

Suppose $G$ is a group and $H$ is a finite normal subgroup of $G$ : $H$ is a normal subgroup of $G$ that is finite as a group. Then, there exists a group $K$ containing $G$ such that $H$ is characteristic in $K$ .

Finite NPC theorem: This states that a normal subgroup of a finite group can be realized as a characteristic subgroup in some finite group containing it.
NPC theorem: This states that any normal subgroup is potentially characteristic.

The proof follows directly by piecing together facts (1) and (2).

@@ Line 1: / Line 1: @@
-{{subgroup property implication|
-stronger = finite normal subgroup|
-weaker = potentially characteristic subgroup}}
 ==Statement==
 Suppose <math>G</math> is a group and <math>H</math> is a [[finite normal subgroup]] of <math>G</math>: <math>H</math> is a [[normal subgroup]] of <math>G</math> that is [[finite group|finite as a group]]. Then, there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is characteristic in <math>K</math>.
-==Definitions used==
+==Related facts==
-===Potentially characteristic subgroup===
+===Stronger facts===
-{{further|[[Potentially characteristic subgroup]]}}
+* [[Finite NPC theorem]]: This states that a normal subgroup of a finite group can be realized as a characteristic subgroup in some finite group containing it.
+* [[NPC theorem]]: This states that ''any'' [[normal subgroup]] is potentially characteristic.
-A subgroup <math>H</math> of a group <math>G</math> is termed a '''potentially characteristic subgroup''' if there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is a [[characteristic subgroup]] of <math>K</math>.
 ==Facts used==
 # [[uses::Finite normal implies amalgam-characteristic]]
-# [[uses::Amalgam-characteristic implies holomorph-characteristic]]
+# [[uses::Amalgam-characteristic implies potentially characteristic]]
 ==Proof==
 The proof follows directly by piecing together facts (1) and (2).